Timeline for Can we state $\sf V=HOD$ using a single ordinal parameter(other than the formula code)?
Current License: CC BY-SA 4.0
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| Jun 26, 2023 at 19:19 | comment | added | Andreas Blass | The reason some people prefer the first version is that it is closer to the original intention of OD as "definable with an ordinal parameter," i.e. $\exists\phi\exists\alpha\,x=\{y:\phi(y,\alpha)\}$. That isn't a definition in ZF, so one has to bring in the additional parameter $\theta$ and invoke the reflection principle. After all that, one notices that, as long as one needs $\theta$ anyway, one can use it to encode the original parameter $\alpha$. That's technically useful, but it's getting rather far from the original idea. | |
| Jun 26, 2023 at 18:54 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Jun 26, 2023 at 18:34 | vote | accept | Zuhair Al-Johar | ||
| Jun 26, 2023 at 18:31 | comment | added | Joel David Hamkins | @ZuhairAl-Johar It often happens in mathematics that when one has two different statements that are equivalent, then one is more useful for some purposes and the other for other purposes. For example, to validate that a given model satisfies V=HOD, the more generous version is often more convenient, since no coding is required. But to prove a consequence of V=HOD, the second version can be more convenient, since it states a seemingly stronger property. I have seen both used in just this way. | |
| Jun 26, 2023 at 18:24 | comment | added | Joel David Hamkins | @Holo Your objection is answered by my second paragraph. But actually, one can get the pair inside $V_\beta$ even without $\beta$ as an element. That is, the model can identify which pair it woud be, and so the $\beta+1$ subterfuge is not actually necessary. | |
| Jun 26, 2023 at 18:23 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Jun 26, 2023 at 18:22 | comment | added | Holo | In Zuhair's definition you don't act any function on $α$, that is you can't use the pairing function on $α$ and use the resulting ordinals as parameters | |
| Jun 26, 2023 at 18:22 | comment | added | Zuhair Al-Johar | Thanks. Why the first version is often used? | |
| Jun 26, 2023 at 18:20 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |